Math 307 G - Spring 2018Midterm 1

April 20, 2018

Name:

Section:

Student ID Number:

• There are 7 pages in total. Make sure your exam contains all these questions.

• There is one bonus problem. The maximal score you can get is 50.

• You are allowed to use a scientific calculator (no graphing calculators and no calculatorsthat have calculus capabilities) and one hand-written 8.5 by 11 inch page of notes.

• You must show your work on all problems. The correct answer with no supporting work may resultin no credit. Put a box around your FINAL ANSWER for each problem and cross outany work that you don’t want to be graded. Give exact answers wherever possible.

• If you need more room, use the backs of the pages and indicate to the grader that you have doneso.

• Raise your hand if you have a question.

• You have 50 minutes to complete the exam. Budget your time wisely.

Problem 1 10

Problem 2 8

Problem 3 8

Problem 4 12

Problem 5 12

Problem 6 (bonus) 3

Total 50

GOOD LUCK!

1

1. (10 pts)

(a) (5 pts) Find the implicit solution to the equation

dy

dx=

1− e−x

y + sin(y).

(b) (5 pts) Use the substitution v = 6x+ y to find the general solution of the equation

dy

dx= 6x+ y

2

2. (8 pts)

(a) (4 pts) Classify all the equilibrium solutions of the equation y′ = −(y−a)(y2− 1)(e−y − e5)2,with a constant a > 1. Justify your answer.

(b) (4 pts) Let y(t) be a solution of the equation in (a) with y(0) = 1.5. If we know limt→+∞

y(t) = 3.

Find the value of the constant a. Justify your answer.

3

3. (8 pts) Consider the following differential equation

dy

dt= t2 + y2, y(0) = 1.

Find the approximate value of y(3) using Euler’s method, with step size h = 1.

4

4. (12 pts) A tank initially contains 100 L of water in which 10 kg of salt is dissolved. Watercontaining 1 kg/L salt flows into the tank at a constant rate of k L/min (k > 0). The mixtureflows out at a rate of k L/min. Assume that the salt is uniformly distributed in the tank.

(a) (4 pts) Let y(t) be the amount of salt in the tank after t minutes. Write a differential equationfor y(t) (with unit kg) in the tank at any time t, and write the initial condition.

(b) (3 pts) Determine limt→+∞

y(t) and justify your answer.

(c) (5 pts) If we know after 20 minutes, the concentration (NOT the mass) of the salt in thewater is (1 + e2) kg/L. Find the value of k.

5

5. (12 pts) Newton’s law of cooling states that the temperature of an object changes at a rateproportional to the difference between the temperature of the object itself and the temperatureof its surroundings.

(a) (4 pts) There is a cup of ice water in a room with ambient temperature Ts, which satisfies attime t, Ts(t) = 70 + e−t sin(t). The initial temperature of the ice water is 30◦F . Assume theabsolute value of the proportionality constant K is 1. Let T (t) be the temperature of theice water after time t. Write a differential equation with an initial value for T .

(b) (5 pts) What is the temperature of the ice water at time t?

(c) (3 pts) Determine limt→+∞

T (t) and justify your answer.

6

6. (3 pts) Bonus Problem

Consider the differential equation

y2 +

!dy

dt

"2

= 1.

Find all solutions and justify your answer.

7